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- W4366342186 abstract "From the results in the literature, the algebraic set of the hyperbola with parameter $n$ defined by $mathcal{B}_{n}(X, Y, Z)_{mid_{xgeq 4n}}= displaystyle lbrace left(X: Y: Zright)in mathbb{P}^{2}(mathbb{Q}) vert displaystyle Y^{2}=X^{2}-4nXZ rbrace$ where $n$ is a semiprime is proved to be in relation with prime factors of $n$. In the affine space over $mathbb{Z}_{geqslant 4n}times mathbb{Z}_{geqslant 0}$, this set has exactly 5 points $displaystylelbrace P_{0}, P_{1}, P_{2}, P_{3}, P_{4} rbrace$ with $P_{2}+P_{3}=P_{1}+2P_{2}=P_{4}$ for which knowledge of $P_{2}$ or $P_{3}$ yields the factorization of $n$. However, The non cyclicity of this group structure over rationals and integers and moreover its non good reduction over finite fields constitute the main difficulty in finding its solutions. In this paper we describe an approach to finding $P_{2}$ and $P_{3}$. We introduce the concept of Hyperbola X-root and Y-root that the solution's greatest common divisors with $n$ reveal prime factors of $n$. We prove that $P_{2}$ and $P_{3}$ can be found on a singular Weierstrass curve isomorphic to a Jacobi quartic using the Hyperbola X-root and Y-root. We present the mathematical framework for this approach." @default.
- W4366342186 created "2023-04-20" @default.
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- W4366342186 date "2023-04-15" @default.
- W4366342186 modified "2023-09-26" @default.
- W4366342186 title "A New Hyperbola based Approach to factoring Integers" @default.
- W4366342186 doi "https://doi.org/10.48550/arxiv.2304.07474" @default.
- W4366342186 hasPublicationYear "2023" @default.
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